Database Backup

Overview
Here you can backup your database, which is highly recommended in the event of a server crash. You should not totally rely on the backup option here, so maybe a backup from within your MySQL management software is also a good idea.
Options > Download File
If you choose this option the 'Save As' dialogue is envoked asking you to save the file somewhere safe. Removeable media such as a flash drive is recommended.
Options > Compress File
For large databases you should select the compression option to make the file size smaller. Compression is via GZip.
Restore Database
Restoring your database is fairly easy. You just need to navigate to your MySQL management software and run the backup SQL schematic. Most servers have PHPMyAdmin or Adminer installed. Alternatively you can use other good database software. Below are some example:

HeidiSQL (My software of choice for local administration on Windows and one of the best free pieces of software out there)
PHPMyAdmin (Installed on many servers as the default database software. PHP based.)
Adminer (A better option than PHPMyAdmin. A single PHP file. Terrific for database management)
SQL Buddy (Lightweight PHP based software. Great alternative to PHPMyAdmin)
MySQL Workbench (Free Windows/Mac/Linux based software from the MySQL creators)
Database Beaver (Free database management software for Windows. Java based.)
Database Browser (Free database management software for Windows)
Sequel Pro (Free database management software for the MAC)